A067024 -id:A067024 - cp's OEIS frontend

A067027 Numbers n such that (prime(n)# + 4)/2 is a prime, where x# is the primorial A034386(x).

1, 2, 3, 4, 6, 10, 11, 12, 15, 17, 29, 48, 63, 77, 88, 187, 190, 338, 1133, 1311, 1832, 2782, 2907, 3180, 3272, 5398, 17530

Offset: 1

Labos Elemer, Dec 29 2001

nonn

Numbers n such that [A002110(n)/2]+2 is prime.
These primes are products of consecutive odd primes plus 2: 2+[3.5.7.....p(n)] if n is here.
a(19)-a(22) are Fermat and Lucas PRPs. (prime(2782)# + 4)/2 has 10865 digits. PFGW Version 1.2.0 for Windows [FFT v23.8] Primality testing (p(2782)#+4)/2 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 5 Running N+1 test using discriminant 13, base 1+sqrt(13) (p(2782)#+4)/2 is Fermat and Lucas PRP! - Jason Earls, Dec 12 2006
a(28) > 25000. - Robert Price, Sep 29 2017

Cf. A002110, A067024, A065026.

p = 1; Do[p = p*Prime[n]; If[PrimeQ[(p + 4)/2], Print[n]], {n, 1, 400} ]
Flatten[Position[FoldList[Times,Prime[Range[3000]]],?(PrimeQ[ (#+4)/2]&)]] (* _Harvey P. Dale, May 24 2015 *)

n=0;pr=1/2;forprime(p=2,1e4,n++;pr*=p;if(ispseudoprime(pr+2),print1(n", "))) \\ Charles R Greathouse IV, Jul 25 2011

More terms from Robert G. Wilson v, Dec 30 2001
a(19)-a(22) from Jason Earls, Dec 12 2006
a(23) from Ray Chandler, Jun 16 2013
a(24)-a(27) from Robert Price, Sep 29 2017

A067026 (Prime(n)# - 4)/2 is prime, where x# is the primorial A034386(x).

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 11, 13, 16, 20, 27, 39, 83, 103, 122, 129, 145, 279, 393, 608, 798, 929, 1164, 1266, 1491, 2043, 3276, 3426, 7119, 15711, 18424

Offset: 1

Labos Elemer, Dec 29 2001

nonn

n such that A002110(n)/2 - 2 is prime.

a(33) > 25000. - Robert Price, Sep 29 2017

Cf. A002110, A067024, A067027.

p = 1; Do[p = p*Prime[n]; If[PrimeQ[(p - 4)/2], Print[n]], {n, 1, 400} ]
Flatten[Position[Rest[FoldList[Times,1,Prime[Range[2100]]]],?(PrimeQ[(#-4)/2]&)]] (* _Harvey P. Dale, Nov 22 2014 *)

More terms from Robert G. Wilson v, Dec 30 2001
a(21)-a(27) from Ray Chandler, Jun 16 2013
a(28)-a(32) from Robert Price, Sep 29 2017

A096177 Primes p such that primorial(p)/2 + 2 is prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 31, 37, 47, 59, 109, 223, 307, 389, 457, 1117, 1151, 2273, 9137, 10753, 15727, 25219, 26459, 29251, 30259, 52901, 194471

Offset: 1

Hugo Pfoertner, Jun 27 2004

a(27) > 172000. - Robert Price, May 10 2019

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 16 2019

			a(3)=7 because primorial(7)/2 + 2 = A070826(4) + 2 = 2*3*5*7/2 + 2 = 107 is prime.

Cf. A070826, A096178 primes of the form primorial(p)/2+2, A096547 primes p such that primorial(p)/2-2 is prime, A067024 smallest p+2 that has n distinct prime factors, A014545 primorial primes, A087398.

k = 1; Do[If[PrimeQ[n], k = k*n; If[PrimeQ[k/2 + 2], Print[n]]], {n, 2, 100000}] (* Ryan Propper, Jul 03 2005 *)

P=1/2;forprime(p=2,1e4,if(isprime((P*=p)+2), print1(p", "))) \\ Charles R Greathouse IV, Mar 14 2011

7 additional terms, corresponding to probable primes, from Ryan Propper, Jul 03 2005
Edited by T. D. Noe, Oct 30 2008
a(26) from Robert Price, May 10 2019
a(27) from Tyler Busby, Mar 17 2024

A087398 Primes of the form primorial(P(k))/2-2.

Original entry on oeis.org

13, 103, 1153, 15013, 255253, 4849843, 111546433, 100280245063, 152125131763603, 16294579238595022363, 278970415063349480483707693, 11992411764462614086353260819346129198103, 481473710367991963528473107950567214598209565303106537707981745633

Offset: 1

Cino Hilliard, Oct 21 2003

nonn

Twinmorial numbers are the partial products of twin primes. Sum of reciprocals = 0.08756985926348207565388288916..

The next term (a(14)) has 174 digits. - Harvey P. Dale, Mar 30 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..19

Cf. A096177 primes k such that primorial(k)/2+2 is prime, A096178 primes of the form primorial(k)/2+2, A096547 Primes k such that primorial(k)/2-2 is prime, A067024 smallest p+2 that has n distinct prime factors, A014545 primorial primes.

Select[#/2-2&/@Rest[FoldList[Times,1,Prime[Range[100]]]],PrimeQ] (* Harvey P. Dale, Mar 30 2013 *)

twimorial(n) = { s=0; p=3; forprime(x=5,n, if(isprime(x-2),c1++); p=p*x; if(isprime(p-2), print1(p-2","); c2++; s+=1.0/(p-2); ) ); print(); print(s) }

v=[];pr=1; forprime(p=3,2357,pr*=p; if(ispseudoprime(pr-2),v=concat(v,pr-2))) \\ Charles R Greathouse IV, Feb 14 2011

Twins 3*5 = 15 = p+2. p=13.

Description corrected by Hugo Pfoertner, Jun 25 2004

One more term (a(13)) added by Harvey P. Dale, Mar 30 2013

A096178 Primes of the form primorial(p)/2+2.

Original entry on oeis.org

3, 5, 17, 107, 15017, 3234846617, 100280245067, 3710369067407, 307444891294245707, 961380175077106319537, 139867498408927468089138080936033904837498617

Offset: 1

Hugo Pfoertner, Jun 27 2004

nonn

Primes of the form A070826(n)+2.

			a(4) = 107 because 107 is a prime of the form primorial(7)/2 + 2 = A070826(4) + 2 = 2*3*5*7/2 + 2.

Amiram Eldar, Table of n, a(n) for n = 1..18

Cf. A070826, A096177 (primorial(p)/2+2 is prime), A096547 (primorial(p)/2-2 is prime), A067024 (smallest p+2 that has n distinct prime factors), A014545 (primorial primes), A087398.

for(n=1,30,p=prod(k=1,n,prime(k))/2+2;if(ispseudoprime(p),print1(p,", "))) \\ Hugo Pfoertner, Dec 26 2019

a(n) = A070826(A096177(n)) + 2. - Amiram Eldar, Dec 26 2019

a(1) inserted by Amiram Eldar, Dec 26 2019

A100277 Primes resulting from A100276.

Original entry on oeis.org

13, 103, 1153, 15013, 255253, 4849843, 111546433, 6581239663, 467268016213, 34110565183693, 2831176910246683, 251974745011954963, 24441550266159631603, 4668336100836489636553, 1573229265981897007519033

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Dec 29 2004

nonn

			3*5-2=13 is prime;
3*5*7-2=103 is prime;
3*5*7*11-2=1153 is prime;
3*5*7*11*13-2=15013 is prime.

Cf. A100276.

Cf. A067024. [From R. J. Mathar, Sep 02 2008]

Corrected and extended by Emeric Deutsch, Mar 26 2005

A098027 Smallest prime p such that p+2 is a product of exactly n distinct primes.

Original entry on oeis.org

3, 13, 103, 1153, 15013, 255253, 4849843, 111546433, 4360010653, 100280245063, 5245694198743, 152125131763603, 7149881192889433, 421842990380476663, 16294579238595022363, 1106494163767990292293, 74135108972455349583763

Offset: 1

Lekraj Beedassy, Sep 10 2004

nonn

In most cases a(n) = pn! - 2, where pn! is "factorial of odd primes": from 3 to n-th odd prime. - Zak Seidov - Zak Seidov and Don Reble, Mar 10 2006

			3+2=5 (1 prime), 13+2=15=3*5 (2 primes), 103+2=105=3*5*7 (3 primes), ...
1153 is the 4th entry in the sequence because it is followed by primes 1783, 1993, 2143, 2803, ... with the property 1153+2 = 3*5*7*11, 1783+2 = 3*5*7*17, 1993+2 = 3*5*7*19, 2143+2 = 3*5*11*13, 2803+2 = 3*5*11*17, ...

Cf. A067024 [From R. J. Mathar, Sep 02 2008]

Extended by Ray Chandler, Sep 18 2004
More terms from Zak Seidov and Don Reble, Mar 10 2006
Corrected by Don Reble, Apr 03 2006

cp's OEIS Frontend

A067027 Numbers n such that (prime(n)# + 4)/2 is a prime, where x# is the primorial A034386(x).

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A067026 (Prime(n)# - 4)/2 is prime, where x# is the primorial A034386(x).

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A096177 Primes p such that primorial(p)/2 + 2 is prime.

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A087398 Primes of the form primorial(P(k))/2-2.

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A096178 Primes of the form primorial(p)/2+2.

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A100277 Primes resulting from A100276.

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A098027 Smallest prime p such that p+2 is a product of exactly n distinct primes.

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